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Theorem ablcom 15467
Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
Hypotheses
Ref Expression
ablcom.b  |-  B  =  ( Base `  G
)
ablcom.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ablcom  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )

Proof of Theorem ablcom
StepHypRef Expression
1 ablcmn 15456 . 2  |-  ( G  e.  Abel  ->  G  e. CMnd
)
2 ablcom.b . . 3  |-  B  =  ( Base `  G
)
3 ablcom.p . . 3  |-  .+  =  ( +g  `  G )
42, 3cmncom 15466 . 2  |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
51, 4syl3an1 1218 1  |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1654    e. wcel 1728   ` cfv 5489  (class class class)co 6117   Basecbs 13507   +g cplusg 13567  CMndccmn 15450   Abelcabel 15451
This theorem is referenced by:  ablinvadd  15472  ablsub2inv  15473  ablsubadd  15474  abladdsub  15477  ablpncan3  15479  ablsub32  15484  eqgabl  15492  subgabl  15493  ablnsg  15500  lsmcomx  15509  divsabl  15518  frgpnabl  15524  ngplcan  18695  r1pid  20120  cnaddcom  29943  toycom  29944  lflsub  30039  lfladdcom  30044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-iota 5453  df-fv 5497  df-ov 6120  df-cmn 15452  df-abl 15453
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